Honeymooners have long laid claim to the dreamy islands of French Polynesia. But Josh Rottenberg finds that rental bungalows on Moorea and Tahiti are giving families their own sweet taste of paradise.

It's not an easy word to translate, but loosely speaking, it means feeling blah, worn out, fed up. You might be fiu at the end of a grinding workweek or the morning after a massive party. Above all, fiu (pronounced "few") means you want to do nothing but relax—in which case you're fortunate if you find yourself in French Polynesia. For decades, travelers here have renewed their spirits with the beautiful beaches, laid-back vibe, and soothing breezes of these South Pacific islands. Spend a little time on a strip of white sand, sipping Hinano beer and staring at an endless expanse of impossibly blue water, and you quickly understand how the place could inspire you to flee your regularly scheduled life.

Like so many before us, my wife and I, nursing our own nagging case of fiu (history will show there was a lot going around in 2009), took our two young daughters to French Polynesia for a course of tropical treatment. We'd gone to Maui for our honeymoon exactly 10 years earlier, but Tahiti and her sister island, Moorea, promised a remote exoticism unlike anything we—let alone our kids—had ever experienced. To heighten the off-the-grid feeling, rather than opt for the traditional resort hotel, we planned to rent a couple of houses on Moorea, a half-hour ferry ride from Tahiti. No room service or concierge or pillow mints. No mindless checking of e-mail or surfing of Facebook. To the extent we could, we'd go native.

It's easier than ever before to make this kind of trip. In light of its romantic-getaway reputation, French Polynesia has been striving to broaden its reach and make inroads with families. I'd heard that, in addition to an ongoing promotion on national airline Air Tahiti Nui—which throws in two free tickets for children under the age of 11 with the purchase of two adult ones—gorgeous houses were increasingly up for rent through websites like vrbo.com (Vacation Rentals By Owner) and homeaway.com. Families who have left behind a life in the States for greener (and bluer) pastures have discovered just how valuable their real estate is and have begun opening up their bungalows to people like us—for under $200 a night, private beach included.

We'd never imagined bringing our kids—3-year-old Julia and 7-year-old Rebecca—to a newlywed destination like French Polynesia. But the prospect of getting a direct, unfiltered experience of the islands, without fear of our darling daughters embarrassing us by doing cannonballs into a hotel swimming pool, couldn't have been more appealing. As we sat on the plane from Los Angeles to Papeete, Tahiti, surrounded by all those fresh-faced honeymooners, we felt secure in the knowledge that paradise would have room enough for all of us.

Gazing at Moorea from a peak on Tahiti in 1835, Charles Darwin described the island as "a picture in a frame" thanks to the barrier reef that perfectly encircles it. Bora-Bora may carry greater mystique for most Americans, going back to the U.S. military presence there in World War II. But despite its proximity to Tahiti, Moorea remains more undeveloped—and, some would argue, more beautiful. The entire 80-square-mile isle, which takes about an hour to circumnavigate and doesn't have a single streetlight, claims less than 800 hotel rooms, fewer than the number of rooms in a single large resort on Oahu's Waikiki Beach. And although the population has more than tripled in the past 25 years, it's still only 16,000.

Arriving at the Bali Hai Boys Beach House outside the tiny village of Maharepa on Moorea's northeastern coast, we were greeted with hugs and fragrant leis by Therese Rio, who more than compensated for her limited English with her kind, Gauguin smile and effusive warmth. Therese's late partner, Hugh Kelley, was one of the original "Bali Hai Boys," a trio of enterprising California friends who came to Moorea in the early 1960s and founded the Club Bali Hai, the first in a string of successful hotels. In 1962, Life magazine ran a profile of the Bali Hai Boys that helped introduce Americans to French Polynesia. "Suddenly, everyone wanted to see the Bali Hai Boys on Moorea," Kelley's son Hiro told me. Kelley—who is credited with coming up with the idea for overwater bungalows, now ubiquitous at island resorts—died in 1998, leaving behind his fellow Bali Hai Boys—Don "Muk" McCallum and Jay Carlisle—as well as 12 children and a two-bedroom, two-bathroom beach house, where we planned to spend our first three nights.

Bali Hai Boys Beach House is redolent with Kelley family history and blurs the line between indoors and out. Old books, board games, and shells collected over the years by the Kelleys populate the living room; the two bathrooms each have their own indoor gardens; and the patio, right off the large living room, looks out over the Pacific Ocean. Behind the house, a picture-postcard white-sand beach with its own weather-beaten private dock would be exclusively ours for the next few days. Poema Kelley, one of Hugh's daughters, suggested the kids might enjoy feeding bread crumbs to fish off the end of our dock. Julia and Rebecca scampered out immediately with a baguette, and sure enough, swarms of colorful tropical creatures quickly gathered.

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